A theoretical model of asymmetric wave ripples.

The time development of ripples under sea waves is investigated by means of the weakly nonlinear stability analysis of a flat sandy bottom subjected to the viscous oscillatory flow that is present in the boundary layer at the bottom of propagating sea waves. Second-order effects in the wave steepness are considered, to take into account the presence of the steady drift generated by the surface waves. Hence, the work of Vittori & Blondeaux (1990 J. Fluid Mech. 218, 19-39 (doi:10.1017/S002211209000091X)) is extended by considering steeper waves and/or less deep waters. As shown by the linear analysis of Blondeaux et al. (2000 Eur. J. Mech. B 19, 285-301 (doi:10.1016/S0997-7546(90)00106-I)), because of the presence of a steady velocity component in the direction of wave propagation, ripples migrate at a constant rate that depends on sediment and wave characteristics. The weakly nonlinear analysis shows that the ripple profile is no longer symmetric with respect to ripple crests and troughs and the symmetry index is computed as a function of the parameters of the problem. In particular, a relationship is determined between the symmetry index and the strength of the steady drift. A fair agreement between model results and laboratory data is obtained, albeit further data and analyses are necessary to determine the behaviour of vortex ripples and to be conclusive.